A stochastic approach to quantum statistics distributions: theoretical derivation and Monte Carlo modelling

Abstract

We present a method aimed at a stochastic derivation of the equilibrium distribution of a classical/quantum ideal gas in the framework of the canonical ensemble. The time evolution of these ideal systems is modelled as a series of transitions from one system microstate to another one and thermal equilibrium is reached via a random walk in the single-particle state space. We look at this dynamic process as a Markov chain satisfying the condition of detailed balance and propose a variant of the Monte Carlo Metropolis algorithm able to take into account indistinguishability of identical quantum particles. Simulations performed on different two-dimensional (2D) systems are revealed to be capable of reproducing the correct trends of the distribution functions and other thermodynamic properties. The simulations allow us to show that, away from the thermodynamic limit, a pseudo-Bose-Einstein condensation occurs for a 2D ideal gas of bosons.

Stochastic simulations for the time evolution of systems which obey generalized statistics: fractional exclusion statistics and Gentile’s statistics

“…The model has at its core a time-dependent Markov chain in the microconfiguration state space, which generalizes previous results from Ref. [5]. In the derivation of the stochastic transition probabilities, a division can be made which separates them into acceptance probabilities, corresponding to transition probabilities in a classical ideal gas (i.e.…”

Equivalence between fractional exclusion statistics and self-consistent mean-field theory in interacting-particle systems in any number of dimensions

“…[32]. If we calculate the α ǫ1,ǫ2 , which connects the species δǫ 2 to the species δǫ 1 , then the following relations have to be satisfied: (26) where δǫ 1 corresponds to the intervals δk 1 and δk ′ 1 on the k axis, whereas δǫ 2 corresponds to the intervals δk 2 and δk (27) like in the case of the CSM in a harmonic trap.…”

“…A stochastic method for the simulation of the time evolution of FES systems was introduced in Ref. [26] as a generalization of a similar method used for Bose and Fermi systems [27], whereas the relatively recent experimental realization of the Fermi degeneracy in cold atomic gases has renewed the interest in the theoretical investigation of non-ideal Fermi systems at low temperatures and their interpretation as ideal FES systems [23,[28][29][30][31].…”

Fractional exclusion statistics in systems with localized states

“…The proposed stochastic method generalizes a previous approach for ideal Fermi and Bose gases [18]. The transition rates include the step and the acceptance probabilities.…”